A pair of flushing devices for introducing machining fluid into the machining gap of a wire-cut electric discharge machine are typically placed on a generally vertical line: one above the workpiece, and one below it. The pair of the flushing devices have respective wire guides therein between which the wire electrode travels generally vertically. Fresh dielectric fluid is forced from nozzles of the upper and lower flushing devices so as to be injected into the machining gap, formed between the workpiece being machined and the traveling wire electrode. This flushing with fresh dielectric fluid cools the wire electrode and workpiece, and also flushes chips produced during machining out of the machining gap. If allowed to remain in the machining gap, chips of conductive material can cause undesirable secondary discharge, which degrades machining accuracy. In addition, flushing constantly replaces the used fluid in the machining gap with fresh dielectric, thus providing faster recovery of the insulating properties of the dielectric fluid, which increases the machining rate. The tips of the flushing device nozzles are usually placed as close as possible to the surface of the workpiece to enable more fluid to be efficiently supplied to the gap. When a first cut with a faster metal removal rate is made in a workpiece, a greater amount of electrical energy is normally supplied to the wire electrode. Under first cut conditions, flushing with a large amount of fluid under high pressure is especially important in order to prevent wire breakage. During a first cut, fluid supplied to the machining area is typically pressurized to about 15-20 kgf/cm.sup.2, and the gap between the tips of the nozzles and the workpiece is maintained at approximately 0.05 mm. Following the first cut, one or more "skim cuts" are normally performed. During skim cuts, a smaller amount of electrical energy is supplied to the wire electrode in order to more precisely machine the workpiece to the desired finish dimensions. A skim cut reduces surface roughness in the kerf created during the first cut. A number of skim cuts may be performed, with the machining energy reduced incrementally from one cut to the next. The dielectric fluid pressure setting used for skim cuts is quite low compared to that used for a first cut (typically on the order of 0.5-1.0 kgf/cm.sup.2). The reason for keeping the fluid pressure low is that greater machining precision is required during skim cuts, and maintaining the fluid pressure at a low value suppresses wire vibration, which improves accuracy. This reduction in fluid pressure is possible because, at the lower machining energy and stock removal rates used for skim cuts, the lower flow rate is sufficient to provide adequate cooling and chip removal. The flow rate and pressure of the dielectric fluid are controlled to keep them at the desired values by flow rate adjustment valves. When making skim cuts, in particular, it is important that the flow rate of the dielectric fluid being injected into the machining gap be accurately maintained at a low value. Expensive flow rate equipment is required to make accurate flow rate adjustment over the broad range of values required for both first cuts and skim cuts, and in actual practice, even with good equipment, it is very difficult to accurately control the flow rate at the low values required for skim cuts. Also, in most cases, as the wire electrode moves along the kerf, there can be wide variations in the rate at which the fluid comes at the wire from the front or back, depending on the path of the kerf that was created during the first cut. These undesirable variations, which are especially large when the wire is near a corner in the path of the kerf, can result in variances in the dimensions of the product being made from the workpiece. It can also cause the surface of the finished product to be left with machining marks in the form of lines running parallel to the direction of travel of the wire electrode between the wire guides. There is a need, then, for a simple flushing device that will enable skim cuts to be performed to precise dimensions.